This equation yields two solutions:
t=3.96size 12{t=3 "." "96"} {} and
t=–1.03size 12{t=3 "." "96"} {} . (It is left as an exercise for the reader to verify these solutions.) The time is
t=3.96ssize 12{t=3 "." "96""s"} {} or
–1.03ssize 12{-1 "." "03""s"} {} . The negative value of time implies an event before the start of motion, and so we discard it. Thus,

t=3.96 s.size 12{t=3 "." "96"" s."} {}

Discussion for (a)

The time for projectile motion is completely determined by the vertical motion. So any projectile that has an initial vertical velocity of 14.3 m/s and lands 20.0 m below its starting altitude will spend 3.96 s in the air.

Solution for (b)

From the information now in hand, we can find the final horizontal and vertical velocities
vxsize 12{v rSub { size 8{x} } } {} and
vysize 12{v rSub { size 8{y} } } {} and combine them to find the total velocity
vsize 12{v} {} and the angle
θ0size 12{θ rSub { size 8{0} } } {} it makes with the horizontal. Of course,
vxsize 12{v rSub { size 8{x} } } {} is constant so we can solve for it at any horizontal location. In this case, we chose the starting point since we know both the initial velocity and initial angle. Therefore:

The negative angle means that the velocity is
50.1ºsize 12{"50" "." 1°} {} below the horizontal. This result is consistent with the fact that the final vertical velocity is negative and hence downward—as you would expect because the final altitude is 20.0 m lower than the initial altitude. (See
[link] .)

One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic. He used it to predict the range of a projectile. On level ground, we define
range to be the horizontal distance
Rsize 12{R} {} traveled by a projectile. Galileo and many others were interested in the range of projectiles primarily for military purposes—such as aiming cannons. However, investigating the range of projectiles can shed light on other interesting phenomena, such as the orbits of satellites around the Earth. Let us consider projectile range further.

Trajectories of projectiles on level ground. (a) The greater the initial speed
v0size 12{v rSub { size 8{0} } } {} , the greater the range for a given initial angle. (b) The effect of initial angle
θ0size 12{θ rSub { size 8{0} } } {} on the range of a projectile with a given initial speed. Note that the range is the same for
15ºsize 12{"15"°} {} and
75ºsize 12{"75°"} {} , although the maximum heights of those paths are different.

How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed
v0size 12{v rSub { size 8{0} } } {} , the greater the range, as shown in
[link] (a). The initial angle
θ0size 12{θ rSub { size 8{0} } } {} also has a dramatic effect on the range, as illustrated in
[link] (b). For a fixed initial speed, such as might be produced by a cannon, the maximum range is obtained with
θ0=45ºsize 12{θ rSub { size 8{0} } = "45º"} {} . This is true only for conditions neglecting air resistance. If air resistance is considered, the maximum angle is approximately
38ºsize 12{"38º"} {} . Interestingly, for every initial angle except
45ºsize 12{"45º"} {} , there are two angles that give the same range—the sum of those angles is
90ºsize 12{"90º"} {} . The range also depends on the value of the acceleration of gravity
gsize 12{g} {} . The lunar astronaut Alan Shepherd was able to drive a golf ball a great distance on the Moon because gravity is weaker there. The range
Rsize 12{R} {} of a projectile on
level ground for which air resistance is negligible is given by

fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.

Tarell

what is the actual application of fullerenes nowadays?

Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

Join the discussion...

what is the Synthesis, properties,and applications of carbon nano chemistry

Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.